On Sets With Special Structures

In this thesis,we mainly research on the set without geometric progressions,maximal and minimal sets.The main results are summarized as follows:1.For k≥3,we call a setG（?）(0,1]of real numbers k-good if G contains no geometric progression of length k with integer ratio r>1.In 2015,Nathanson and O’Bryant showed there exists a unique sequence of integers ｛1=A₁^（k）<A₂^（k）<···｝ such that G^（k）=（?）(1/A_2i^（k）,A_2i-1^（k）] is a k-good set.They also determined the values of A₂^（k）,A₃^（k）and A₄^（k）.We further determine the value of A₆^（k）based on the value of A₅^（k）obtained by Fang:Theorem.Let k≥3 be an integer.（ⅰ）If there is no integral power of 3 between2^k-1 and 2^k,then A₆^（k）=2^k 3^k-l,where l is the smallest integer such that 3^l>2^k.（ⅱ）If there is a positive integer l such that2^k-1<3^l<2^k and there is no integral power of 4 between4·3^k-l-1and 2·3^k-l,then（ⅲ）If there is a positive integer l such that2^k-1<3^l<2^k and there is a positive integer m such that4·3^k-l-1<4^m<2·3^k-l,then A₆^（4）=216 and2.A set A of positive integers is called 3-free if it contains no 3-term arithmetic progression.Furthermore,such A is called maximal if it is not properly contained in any other3-free set.In 2006,S.Savchev and F.Chen proved that there exists a maximal 3-free set｛a₁<a₂<···｝of positive integers with the property that （?）(a_n+1-a_n)=∞.In this thesis the following result is proved:Theorem.For any given ε with 0<ε<2^1/2-1 and any 3-free set A=｛a₁<a₂<···｝of positive integers with a_n+1≥2a_n for all positive integers n≥n₀,there exists a set B=｛b₁<b₂<··｝of positive integers such that:（ⅰ）A（?）B;（ⅱ）B is maximal 3-free;（ⅲ）b_n+1≥(2^1/2-ε)b_nfor all positive integersn≥n₀.For a positive integer h≥2,the set A of nonnegative integers is called an asymptotic basis of order h if every sufficiently large integer can be represented as a sum of h elements of A.Furthermore,such A is defined as minimal if no proper subset of A has this property.For a set A of nonnegative integers,letA（x）=card｛a（?）A:1≤a≤x｝.Denote r_A（n）by the number of representations of n of the form n=a+a’,where a≤a’and a,a’（?）A.In 2010,Jańczak and Schoen proved that there exists a minimal asymptotic basis A of order 2 such that for every positive integer k,1/2k≤A（k）≤1/2k+1.In this thesis,we generalize the above result and obtain that:Theorem.Let （?） be any given set of positive integers such that t₀≥7 and t_n+1≥3t_n+6 for all nonnegative integers n.Then there exists a minimal asymptotic basis A of order 2 such that:（ⅰ）1/2k≤A（k）≤1/2k+1 for every positive integer k;（ⅱ）r_A（T _n）=1 for every positive integer n,whereT_n=t_n-2 for odd t_n and T_n=t_n-1 for even t_n.